# HG changeset patch # User Scott Morrison # Date 1275458862 25200 # Node ID 121c580d5ef73b24f799f09073982220e911bb0e # Parent 7cb7de37cbf9b94d07692ddf151e345dad4eaade editting all over the place diff -r 7cb7de37cbf9 -r 121c580d5ef7 text/comm_alg.tex --- a/text/comm_alg.tex Tue Jun 01 21:44:09 2010 -0700 +++ b/text/comm_alg.tex Tue Jun 01 23:07:42 2010 -0700 @@ -109,8 +109,8 @@ \nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} Let us check this directly. -According to \cite[3.2.2]{MR1600246}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. -\nn{say something about $t$-degree? is this in Loday?} +The algebra $k[t]$ has Koszul resolution $k[t] \tensor k[t] \xrightarrow{t\tensor 1 - 1 \tensor t} k[t] \tensor k[t]$, which has coinvariants $k[t] \xrightarrow{0} k[t]$. This is equal to its homology, so we have $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and zero for $i\ge 2$. +(See also \cite[3.2.2]{MR1600246}.) This computation also tells us the $t$-gradings: $HH_0(k[t]) \iso k[t]$ is in the usual grading, and $HH_1(k[t]) \iso k[t]$ is shifted up by one. We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. The fixed points of this flow are the equally spaced configurations. @@ -123,9 +123,9 @@ of course $\Sigma^0(S^1)$ is a point. Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ and is zero for $i\ge 2$. -\nn{say something about $t$-degrees also matching up?} +Note that the $j$-grading here matches with the $t$-grading on the algebraic side. -By xxxx and \ref{ktchprop}, +By xxxx and Proposition \ref{ktchprop}, the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree diff -r 7cb7de37cbf9 -r 121c580d5ef7 text/deligne.tex --- a/text/deligne.tex Tue Jun 01 21:44:09 2010 -0700 +++ b/text/deligne.tex Tue Jun 01 23:07:42 2010 -0700 @@ -11,7 +11,7 @@ (Proposition \ref{prop:deligne} below). Then we sketch the proof. -\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} +\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite[\S2.5]{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} %from http://www.ams.org/mathscinet-getitem?mr=1805894 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. diff -r 7cb7de37cbf9 -r 121c580d5ef7 text/intro.tex --- a/text/intro.tex Tue Jun 01 21:44:09 2010 -0700 +++ b/text/intro.tex Tue Jun 01 23:07:42 2010 -0700 @@ -24,12 +24,12 @@ The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. -Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. +Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. -In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below. +In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below. The relationship between all these ideas is sketched in Figure \ref{fig:outline}. @@ -48,11 +48,11 @@ \newcommand{\yc}{6} \node[box] at (\xa,\ya) (C) {$\cC$ \\ a topological \\ $n$-category}; -\node[box] at (\xb,\ya) (A) {$A(M; \cC)$ \\ the (dual) TQFT \\ Hilbert space}; +\node[box] at (\xb,\ya) (A) {$\underrightarrow{\cC}(M)$ \\ the (dual) TQFT \\ Hilbert space}; \node[box] at (\xa,\yb) (FU) {$(\cF, \cU)$ \\ fields and\\ local relations}; \node[box] at (\xb,\yb) (BC) {$\bc_*(M; \cC)$ \\ the blob complex}; \node[box] at (\xa,\yc) (Cs) {$\cC_*$ \\ an $A_\infty$ \\$n$-category}; -\node[box] at (\xb,\yc) (BCs) {$\bc_*(M; \cC_*)$}; +\node[box] at (\xb,\yc) (BCs) {$\underrightarrow{\cC_*}(M)$}; @@ -77,7 +77,7 @@ \label{fig:outline} \end{figure} -Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. +Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$ and the `small blob complex', and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. \nn{some more things to cover in the intro} @@ -348,7 +348,7 @@ \subsection{Thanks and acknowledgements} -We'd like to thank David Ben-Zvi, Kevin Costello, +We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. diff -r 7cb7de37cbf9 -r 121c580d5ef7 text/ncat.tex --- a/text/ncat.tex Tue Jun 01 21:44:09 2010 -0700 +++ b/text/ncat.tex Tue Jun 01 23:07:42 2010 -0700 @@ -74,7 +74,7 @@ We will concentrate on the case of PL unoriented manifolds. (The ambitious reader may want to keep in mind two other classes of balls. -The first is balls equipped with a map to some other space $Y$. \todo{cite something of Teichner's?} +The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). This will be used below to describe the blob complex of a fiber bundle with base space $Y$. The second is balls equipped with a section of the the tangent bundle, or the frame @@ -86,7 +86,7 @@ of morphisms). The 0-sphere is unusual among spheres in that it is disconnected. Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. -(Actually, this is only true in the oriented case, with 1-morphsims parameterized +(Actually, this is only true in the oriented case, with 1-morphisms parameterized by oriented 1-balls.) For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place. @@ -123,7 +123,7 @@ Most of the examples of $n$-categories we are interested in are enriched in the following sense. The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and -all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category +all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category (e.g.\ vector spaces, or modules over some ring, or chain complexes), and all the structure maps of the $n$-category should be compatible with the auxiliary category structure. @@ -142,7 +142,7 @@ equipped with an orientation of its once-stabilized tangent bundle. Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of their $k$ times stabilized tangent bundles. -(cf. [Stolz and Teichner].) +(cf. \cite{MR2079378}.) Probably should also have a framing of the stabilized dimensions in order to indicate which side the bounded manifold is on. For the moment just stick with unoriented manifolds.} @@ -780,23 +780,6 @@ (actions of homeomorphisms); define $k$-cat $\cC(\cdot\times W)$} -\nn{need to revise stuff below, since we no longer have the sphere axiom} - -Recall that Axiom \ref{axiom:spheres} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. - -\begin{lem} -For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ -\end{lem} - -\begin{lem} -For a topological $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) = \cC(B).$$ -\end{lem} - -\begin{lem} -For an $A_\infty$ $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) \quism \cC(B).$$ -\end{lem} - - \subsection{Modules} Next we define plain and $A_\infty$ $n$-category modules. diff -r 7cb7de37cbf9 -r 121c580d5ef7 text/tqftreview.tex --- a/text/tqftreview.tex Tue Jun 01 21:44:09 2010 -0700 +++ b/text/tqftreview.tex Tue Jun 01 23:07:42 2010 -0700 @@ -4,8 +4,8 @@ \label{sec:fields} \label{sec:tqftsviafields} -In this section we review the construction of TQFTs from ``topological fields". -For more details see \cite{kw:tqft}. +In this section we review the notion of a ``system of fields and local relations". +For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-TQFT}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 submanifold of $X$, then $X \setmin Y$ implicitly means the closure @@ -21,18 +21,17 @@ oriented, topological, smooth, spin, etc. --- but for definiteness we will stick with unoriented PL.) -%Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. +Fix a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. A $n$-dimensional {\it system of fields} in $\cS$ is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ together with some additional data and satisfying some additional conditions, all specified below. -Before finishing the definition of fields, we give two motivating examples -(actually, families of examples) of systems of fields. +Before finishing the definition of fields, we give two motivating examples of systems of fields. \begin{example} \label{ex:maps-to-a-space(fields)} -Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps +Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps from X to $B$. \end{example} @@ -42,7 +41,7 @@ the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by $j$-morphisms of $C$. One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. -This is described in more detail below. +This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. \end{example} Now for the rest of the definition of system of fields. @@ -144,6 +143,47 @@ \nn{remark that if top dimensional fields are not already linear then we will soon linearize them(?)} +For top dimensional ($n$-dimensional) manifolds, we're actually interested +in the linearized space of fields. +By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is +the vector space of finite +linear combinations of fields on $X$. +If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. +Thus the restriction (to boundary) maps are well defined because we never +take linear combinations of fields with differing boundary conditions. + +In some cases we don't linearize the default way; instead we take the +spaces $\lf(X; a)$ to be part of the data for the system of fields. +In particular, for fields based on linear $n$-category pictures we linearize as follows. +Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by +obvious relations on 0-cell labels. +More specifically, let $L$ be a cell decomposition of $X$ +and let $p$ be a 0-cell of $L$. +Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that +$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. +Then the subspace $K$ is generated by things of the form +$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader +to infer the meaning of $\alpha_{\lambda c + d}$. +Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. + +\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; +will do something similar below; in general, whenever a label lives in a linear +space we do something like this; ? say something about tensor +product of all the linear label spaces? Yes:} + +For top dimensional ($n$-dimensional) manifolds, we linearize as follows. +Define an ``almost-field" to be a field without labels on the 0-cells. +(Recall that 0-cells are labeled by $n$-morphisms.) +To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism +space determined by the labeling of the link of the 0-cell. +(If the 0-cell were labeled, the label would live in this space.) +We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). +We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the +above tensor products. + + +\subsection{Systems of fields from $n$-categories} +\label{sec:example:traditional-n-categories(fields)} We now describe in more detail systems of fields coming from sub-cell-complexes labeled by $n$-category morphisms. @@ -226,43 +266,6 @@ \medskip -For top dimensional ($n$-dimensional) manifolds, we're actually interested -in the linearized space of fields. -By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is -the vector space of finite -linear combinations of fields on $X$. -If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. -Thus the restriction (to boundary) maps are well defined because we never -take linear combinations of fields with differing boundary conditions. - -In some cases we don't linearize the default way; instead we take the -spaces $\lf(X; a)$ to be part of the data for the system of fields. -In particular, for fields based on linear $n$-category pictures we linearize as follows. -Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by -obvious relations on 0-cell labels. -More specifically, let $L$ be a cell decomposition of $X$ -and let $p$ be a 0-cell of $L$. -Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that -$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. -Then the subspace $K$ is generated by things of the form -$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader -to infer the meaning of $\alpha_{\lambda c + d}$. -Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. - -\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; -will do something similar below; in general, whenever a label lives in a linear -space we do something like this; ? say something about tensor -product of all the linear label spaces? Yes:} - -For top dimensional ($n$-dimensional) manifolds, we linearize as follows. -Define an ``almost-field" to be a field without labels on the 0-cells. -(Recall that 0-cells are labeled by $n$-morphisms.) -To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism -space determined by the labeling of the link of the 0-cell. -(If the 0-cell were labeled, the label would live in this space.) -We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). -We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the -above tensor products.